Answer to Question #144031 in Abstract Algebra for Jayshree

Question

Express following permutation in S6 in cyclic notation a=(1 2 3 4 5 6
3 4 1 5 6 2)

Answer 1

The element of S6 defined by f(1) = 3, f(2) = 4, f(3) = 1, f(4) = 5, f(5) = 6 and f(6) = 2 has the following representation by two-line notation: 

$\begin{pmatrix} 1&2&3&4&5&6\\ 3&4&1&5&6&2 \end{pmatrix}$ = (1 3) (2 4 5 6)

Explanation : Step 1:

we start with 1.1 goes to 3 and 3 goes back to 1 again. so first cycle is (1,3).

Step 2: we start second cycle with 2. 2 goes to 4, 4 goes to 5,5 goes to 6, 6 goes to 2 again. so second cycle is (2 4 5 6)